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Theorem dfon4 32328
 Description: Another quantifier-free definition of On. (Contributed by Scott Fenton, 4-May-2014.)
Assertion
Ref Expression
dfon4 On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))

Proof of Theorem dfon4
StepHypRef Expression
1 dfon3 32327 . 2 On = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
2 df-ima 5280 . . . 4 (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∖ ( I ∪ E )) ↾ Trans )
3 df-res 5279 . . . . . 6 (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V))
4 indif1 4015 . . . . . 6 (( SSet ∖ ( I ∪ E )) ∩ ( Trans × V)) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
53, 4eqtri 2783 . . . . 5 (( SSet ∖ ( I ∪ E )) ↾ Trans ) = (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
65rneqi 5508 . . . 4 ran (( SSet ∖ ( I ∪ E )) ↾ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
72, 6eqtri 2783 . . 3 (( SSet ∖ ( I ∪ E )) “ Trans ) = ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E ))
87difeq2i 3869 . 2 (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans )) = (V ∖ ran (( SSet ∩ ( Trans × V)) ∖ ( I ∪ E )))
91, 8eqtr4i 2786 1 On = (V ∖ (( SSet ∖ ( I ∪ E )) “ Trans ))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632  Vcvv 3341   ∖ cdif 3713   ∪ cun 3714   ∩ cin 3715   I cid 5174   E cep 5179   × cxp 5265  ran crn 5268   ↾ cres 5269   “ cima 5270  Oncon0 5885   SSet csset 32267   Trans ctrans 32268 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pow 4993  ax-pr 5056  ax-un 7116 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-int 4629  df-iun 4675  df-br 4806  df-opab 4866  df-mpt 4883  df-tr 4906  df-id 5175  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-xp 5273  df-rel 5274  df-cnv 5275  df-co 5276  df-dm 5277  df-rn 5278  df-res 5279  df-ima 5280  df-ord 5888  df-on 5889  df-suc 5891  df-iota 6013  df-fun 6052  df-fn 6053  df-f 6054  df-fo 6056  df-fv 6058  df-1st 7335  df-2nd 7336  df-txp 32289  df-sset 32291  df-trans 32292 This theorem is referenced by: (None)
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